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# Vehicular and pedestrian traffic models: from flow forecast to safety management

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We study the derivation of macroscopic traffic models out of optimal speed and follow-the-leader particle dynamics as hydrodynamic limits of non-local Povzner-type kinetic equations. As a first step, we show that optimal speed vehicle dynamics produce a first order macroscopic model with non-local flux. Next, we show that non-local follow-the-leader vehicle dynamics have a universal macroscopic counterpart in the second order Aw-Rascle-Zhang traffic model, at least when the non-locality of the interactions is sufficiently small. Finally, we show that the same qualitative result holds also for a general class of follow-the-leader dynamics based on the headway of the vehicles rather than on their speed. We also investigate the correspondence between the solutions to particle models and their macroscopic limits by means of numerical simulations.

We study the derivation of non-local macroscopic traffic models out of optimal speed and follow-the-leader particle dynamics as hydrodynamic limits of non-local Povzner-type kinetic equations. As a first step, we show that optimal speed vehicle dynamics produce a first order macroscopic model with non-local flux. Next, we show that non-local follow-the-leader vehicle dynamics have a universal macroscopic counterpart in the second order Aw-Rascle-Zhang traffic model, at least when the non-locality of the interactions is sufficiently small. Finally, we show that the same qualitative result holds also for a general class of follow-the-leader dynamics based on the headway of the vehicles rather than on their speed. We also investigate the correspondence between the solutions to particle models and their macroscopic limits by means of numerical simulations.

We present the derivation of macroscopic traffic models from car-following vehicle dynamics by means of hydrodynamic limits of an Enskog-type kinetic description. We consider the superposition of Follow-the-Leader (FTL) interactions and relaxation towards a traffic-dependent Optimal Velocity (OV) and we show that the resulting macroscopic models depend on the relative frequency between these two microscopic processes. If FTL interactions dominate then one gets an inhomogeneous Aw-Rascle-Zhang model, whose (pseudo) pressure and stability of the uniform flow are precisely defined by some features of the microscopic FTL and OV dynamics. Conversely, if the rate of OV relaxation is comparable to that of FTL interactions then one gets a Lighthill-Whitham-Richards model ruled only by the OV function. Unlike other formally analogous results, our approach builds the macroscopic models as physical limits of particle dynamics rather than simply assessing the convergence of microscopic to macroscopic solutions under suitable numerical discretisations.

In this paper, we derive second order hydrodynamic traffic models from kinetic-controlled equations for driver-assist vehicles. At the vehicle level we take into account two main control strategies synthesising the action of adaptive cruise controls and cooperative adaptive cruise controls. The resulting macroscopic dynamics fulfil the anisotropy condition introduced in the celebrated Aw-Rascle-Zhang model. Unlike other models based on heuristic arguments, our approach unveils the main physical aspects behind frequently used hydrodynamic traffic models and justifies the structure of the resulting macroscopic equations incorporating driver-assist vehicles. Numerical insights show that the presence of driver-assist vehicles produces an aggregate homogenisation of the mean flow speed, which may also be steered towards a suitable desired speed in such a way that optimal flows and traffic stabilisation are reached.

We present the derivation of Generic Second Order macroscopic Models (GSOMs) of vehicular traffic out of a Follow-the-Leader particle description via a kinetic approach. In the vehicle interactions, we introduce a binary control modelling the automatic feedback provided by driver-assist vehicles, then we upscale such a controlled particle dynamics by means of an Enskog-based hydrodynamic limit. The resulting macroscopic model contains in turn a control term inherited from the microscopic interactions. We show that such a control may be chosen so as to optimise global traffic trends, such as the vehicle flux or the road congestion, constrained by the GSOM dynamics. By means of numerical simulations, we investigate the effect of this control hierarchy in some specific case studies, which exemplify the multiscale path from the vehicle-wise implementation of a driver-assist control to its optimal hydrodynamic design.

We study the derivation of macroscopic traffic models from car-following vehicle dynamics by means of hydrodynamic limits of an Enskog-type kinetic description. We consider the superposition of Follow-the-Leader (FTL) interactions and relaxation towards a traffic-dependent Optimal Velocity (OV) and we show that the resulting macroscopic models depend on the relative frequency between these two microscopic processes. If FTL interactions dominate then one gets an inhomogeneous Aw-Rascle-Zhang model, whose (pseudo) pressure and stability of the uniform flow are precisely defined by some features of the microscopic FTL and OV dynamics. Conversely, if the rate of OV relaxation is comparable to that of FTL interactions then one gets a Lighthill-Whitham-Richards model ruled only by the OV function. We further confirm these findings by means of numerical simulations of the particle system and the macroscopic models. Unlike other formally analogous results, our approach builds the macroscopic models as physical limits of particle dynamics rather than assessing the convergence of microscopic to macroscopic solutions under suitable numerical discretisations.

We study the derivation of macroscopic traffic models from car-following vehicle dynamics by means of hydrodynamic limits of an Enskog-type kinetic description. We consider the superposition of Follow-the-Leader (FTL) interactions and relaxation towards a traffic-dependent Optimal Velocity (OV) and we show that the resulting macroscopic models depend on the relative frequency between these two microscopic processes. If FTL interactions dominate then one gets an inhomogeneous Aw-Rascle-Zhang model, whose (pseudo) pressure and stability of the uniform flow are precisely defined by some features of the microscopic FTL and OV dynamics. Conversely, if the rate of OV relaxation is comparable to that of FTL interactions then one gets a Lighthill-Whitham-Richards model ruled only by the OV function. We further confirm these findings by means of numerical simulations of the particle system and the macroscopic models. Unlike other formally analogous results, our approach builds the macroscopic models as physical limits of particle dynamics rather than assessing the convergence of microscopic to macroscopic solutions under suitable numerical discretisations.

We study the derivation of generic high order macroscopic traffic models from a follow-the-leader particle description via a kinetic approach. First, we recover a third order traffic model as the hydrodynamic limit of an Enskog-type kinetic equation. Next, we introduce in the vehicle interactions a binary control modelling the automatic feedback provided by driver-assist vehicles and we upscale such a new particle description by means of another Enskog-based hydrodynamic limit. The resulting macroscopic model is now a Generic Second Order Model (GSOM), which contains in turn a control term inherited from the microscopic interactions. We show that such a control may be chosen so as to optimise global traffic trends, such as the vehicle flux or the road congestion, constrained by the GSOM dynamics. By means of numerical simulations, we investigate the effect of this control hierarchy in some specific case studies, which exemplify the multiscale path from the vehicle-wise implementation of a driver-assist control to its optimal hydrodynamic design.

In this paper, we derive second order hydrodynamic traffic models from kinetic-controlled equations for driver-assist vehicles. At the vehicle level we take into account two main control strategies synthesising the action of adaptive cruise controls and cooperative adaptive cruise controls. The resulting macroscopic dynamics fulfil the anisotropy condition introduced in the celebrated Aw-Rascle-Zhang model. Unlike other models based on heuristic arguments, our approach unveils the main physical aspects behind frequently used hydrodynamic traffic models and justifies the structure of the resulting macroscopic equations incorporating driver-assist vehicles. Numerical insights show that the presence of driver-assist vehicles produces an aggregate homogenisation of the mean flow speed, which may also be steered towards a suitable desired speed in such a way that optimal flows and traffic stabilisation are reached.

We study the derivation of generic high order macroscopic traffic models from a follow-the-leader particle description via a kinetic approach. First, we recover a third order traffic model as the hydrodynamic limit of an Enskog-type kinetic equation. Next, we introduce in the vehicle interactions a binary control modelling the automatic feedback provided by driver-assist vehicles and we upscale such a new particle description by means of another Enskog-based hydrodynamic limit. The resulting macroscopic model is now a Generic Second Order Model (GSOM), which contains in turn a control term inherited from the microscopic interactions. We show that such a control may be chosen so as to optimise global traffic trends, such as the vehicle flux or the road congestion, constrained by the GSOM dynamics. By means of numerical simulations, we investigate the effect of this control hierarchy in some specific case studies, which exemplify the multiscale path from the vehicle-wise implementation of a driver-assist control to its optimal hydrodynamic design.

In this paper, we consider a kinetic description of follow-the-leader traffic models, which we use to study the effect of vehicle-wise driver-assist control strategies at various scales, from that of the local traffic up to that of the macroscopic stream of vehicles. We provide theoretical evidence of the fact that some typical control strategies, such as the alignment of the speeds and the optimisation of the time headways, impact on the local traffic features (for instance, the speed and headway dispersion responsible for local traffic instabilities) but have virtually no effect on the observable macroscopic traffic trends (for instance, the flux/throughput of vehicles). This unobvious conclusion, which is in very nice agreement with recent field studies on autonomous vehicles, suggests that the kinetic approach may be a valid tool for an organic multiscale investigation and possibly the design of driver-assist algorithms.

In this paper we consider a Boltzmann-type kinetic description of Follow-the-Leader traffic dynamics and we study the resulting asymptotic distributions, namely the counterpart of the Maxwellian distribution of the classical kinetic theory. In the Boltzmann-type equation we include a non-constant collision kernel, in the form of a cutoff, in order to exclude from the statistical model possibly unphysical interactions. In spite of the increased analytical difficulty caused by this further non-linearity, we show that a careful application of the quasi-invariant limit (an asymptotic procedure reminiscent of the grazing collision limit) successfully leads to a Fokker-Planck approximation of the original Boltzmann-type equation, whence stationary distributions can be explicitly computed. Our analytical results justify, from a genuinely model-based point of view, some empirical results found in the literature by interpolation of experimental data.

In this paper, we propose a kinetic model of traffic flow with uncertain binary interactions, which explains the scattering of the fundamental diagram in terms of the macroscopic variability of aggregate quantities, such as the mean speed and the flux of the vehicles, produced by the microscopic uncertainty. Moreover, we design control strategies at the level of the microscopic interactions among the vehicles, by which we prove that it is possible to dampen the propagation of such an uncertainty across the scales. Our analytical and numerical results suggest that the aggregate traffic flow may be made more ordered, hence predictable, by implementing such control protocols in driver-assist vehicles. Remarkably, they also provide a precise relationship between a measure of the macroscopic damping of the uncertainty and the penetration rate of the driver-assist technology in the traffic stream.

The kinetic description of vehicular traffic is one of the first examples in which methods of the statistical physics were applied to a particle system different from a standard gas. Such an approach was initiated by the Russian physicist Ilya Prigogine in the sixties, in an attempt to explain the emergence of collective properties as a result of individual ones in systems composed by human beings instead of molecules. Kinetic traffic models constitute nowadays a promising research line in the context of the multiscale aspects of Artificial Intelligence. Indeed, the coexistence of human and computer-based assistance is currently one of the main goals in vehicle dynamics, due to its potential ability to correct the sub-optimal behaviour of individual drivers and to produce beneficial impacts on traffic flow and road safety at larger scales. As an example, we mention the Advanced Driver-Assistance Systems (ADAS), which constitute a real interface between human drivers and machine-based decision making. In this talk, we will review the application of some classical mathematical methods of the kinetic theory, such as e.g., Boltzmann-type collisional equations and Fokker-Planck asymptotics, to these emerging topics in vehicular traffic modelling.

In this paper we consider a Boltzmann-type kinetic description of Follow-the-Leader traffic dynamics and we study the resulting asymptotic distributions, namely the counterpart of the Maxwellian distribution of the classical kinetic theory. In the Boltzmann-type equation we include a non-Maxwellian, viz. non-constant, collision kernel in order to exclude from the statistical model possibly unphysical interactions. In spite of the increased analytical difficulty caused by this further non-linearity, we show that a careful application of the quasi-invariant limit (an asymptotic procedure reminiscent of the grazing collision limit) successfully leads to a Fokker-Planck approximation of the original Boltzmann-type equation, whence stationary distributions can be explicitly computed. Our analytical results justify, from a genuinely model-based point of view, some empirical results found in the literature by interpolation of experimental data.

In this work we investigate the ability of a kinetic approach for traffic dynamics to predict speed distributions obtained through rough data. The present approach adopts the formalism of uncertainty quantification, since reaction strengths are uncertain and linked to different types of driver behaviour or different classes of vehicles present in the flow. Therefore, the calibration of the expected speed distribution has to face the reconstruction of the distribution of the uncertainty. We adopt experimental microscopic measurements recorded on a German motorway, whose speed distribution shows a multimodal trend. The calibration is performed by extrapolating the uncertainty parameters of the kinetic distribution via a constrained optimisation approach. The results confirm the validity of the theoretical setup.

In this paper we consider a kinetic description of follow-the-leader traffic models, which we use to study the effect of vehicle-wise driver-assist control strategies at various scales, from that of the local traffic up to that of the macroscopic stream of vehicles. We provide a theoretical evidence of the fact that some typical control strategies, such as the alignment of the speeds and the optimisation of the time headways, impact on the local traffic features (for instance, the speed and headway dispersion responsible for local traffic instabilities) but have virtually no effect on the observable macroscopic traffic trends (for instance, the flux/throughput of vehicles). This unobvious conclusion, which is in very nice agreement with recent field studies on autonomous vehicles, suggests that the kinetic approach may be a valid tool for an organic multiscale investigation and possibly design of driver-assist algorithms.

We study the derivation of second order macroscopic traffic models from kinetic descriptions. In particular, we recover the celebrated Aw-Rascle model as the hydrodynamic limit of an Enskog-type kinetic equation out of a precise characterisation of the microscopic binary interactions among the vehicles. Unlike other derivations available in the literature, our approach unveils the multiscale physics behind the Aw-Rascle model. This further allows us to generalise it to a new class of second order macroscopic models complying with the Aw-Rascle consistency condition, namely the fact that no wave should travel faster than the mean traffic flow.

The modern mathematical theory of vehicular traffic was born approximately at the beginning of the sixties with some works proposing models of the flow of vehicles along a road at the three main representation scales: the microscopic one, which describes the vehicles as interacting particles; the macroscopic one, which assimilates the vehicles to a continuum with density; the mesoscopic one, which, inspired by the Boltzmann kinetic theory, studies the statistical distribution of the microscopic speeds of the vehicles. In the subsequent years, microscopic and macroscopic models were intensely studied, and quite complete and refined mathematical theories were developed. On the contrary, kinetic models were rediscovered only towards the mid-nineties, parallelly to the initiation of the studies on multi-agent systems.
In this seminar, we show that vehicular traffic is, to all intents and purposes, an unexpected field of application of the classical kinetic theory and that the kinetic approach provides a sound framework to study a large variety of topics ranging from the classical fundamental diagrams to the nowadays rising driver-assist technologies.

We introduce a kinetic description of control problems for vehicular traffic aimed at dampening some structural uncertainties responsible for scattered aggregate trends. In more detail, we model stochastic microscopic interactions among the vehicles, subject to an instantaneous control when they involve driver-assist vehicles. Then, we upscale them to the global flow via a kinetic Boltzmann-type equation. Our approach promotes the idea that multi-agent systems need to be controlled via bottom-up rather than via top-down strategies.

We study the derivation of second order macroscopic traffic models from kinetic descriptions. In particular, we recover the celebrated Aw-Rascle model as the hydrodynamic limit of an Enskog-type kinetic equation out of a precise characterisation of the microscopic binary interactions among the vehicles. Unlike other derivations available in the literature, our approach unveils the multiscale physics behind the Aw-Rascle model. This further allows us to generalise it to a new class of second order macroscopic models complying with the Aw-Rascle consistency condition, namely the fact that no wave should travel faster than the mean traffic flow.

In this talk, we present a hierarchical description of control problems for vehicular traffic, which aim to mitigate speed-dependent risk factors. In particular, we implement mathematically the idea that a few automated cars can be controlled in order to align the speeds in the traffic stream either to each other or to some recommended optimal speed.
We discuss the modelling of stochastic microscopic binary interactions among the vehicles, including a probabilistic description of the penetration rate of automated vehicles. When the interactions involve one of such vehicles, they are further subject to a binary control problem, which aims to reduce the speed gap either with the leading vehicle or with a prescribed congestion-dependent speed. Then, we upscale the interaction rules to the global flow by means of a kinetic Boltzmann-type equation, which we use to investigate the impact of the microscopic control on the macroscopic flow.

In this paper, we propose a kinetic model of traffic flow with uncertain binary interactions, which explains the scattering of the fundamental diagram in terms of the macroscopic variability of aggregate quantities, such as the mean speed and the flux of the vehicles, produced by the microscopic uncertainty. Moreover, we design control strategies at the level of the microscopic interactions among the vehicles, by which we prove that it is possible to dampen the propagation of such an uncertainty across the scales. Our analytical and numerical results suggest that the aggregate traffic flow may be made more ordered, hence predictable, by implementing such control protocols in driver-assist vehicles. Remarkably, they also provide a precise relationship between a measure of the macroscopic damping of the uncertainty and the penetration rate of the driver-assist technology in the traffic stream.

In this paper we formulate a theory of measure-valued linear transport equations on networks. The building block of our approach is the initial/boundary-value problem for the measure-valued linear transport equation on a bounded interval, which is the prototype of an arc of the network. For this problem we give an explicit representation formula of the solution, which also considers the total mass flowing out of the interval. Then we construct the global solution on the network by gluing all the measure-valued solutions on the arcs by means of appropriate distribution rules at the vertexes. The measure-valued approach makes our framework suitable to deal with multiscale flows on networks, with the microscopic and macroscopic phases represented by Lebesgue-singular and Lebesgue-absolutely continuous measures, respectively, in time and space.

In this paper we introduce and discuss numerical schemes for the approximation of kinetic equations for flocking behavior with phase transitions that incorporate uncertain quantities. This class of schemes here considered make use of a Monte Carlo approach in the phase space coupled with a stochastic Galerkin expansion in the random space. The proposed methods naturally preserve the positivity of the statistical moments of the solution and are capable to achieve high accuracy in the random space. Several tests on a kinetic alignment model with self propulsion validate the proposed methods both in the homogeneous and inhomogeneous setting, shading light on the influence of uncertainties in phase transition phenomena driven by noise such as their smoothing and confidence bands.

This paper is devoted to the construction of structure preserving stochastic Galerkin schemes for Fokker-Planck type equations with uncertainties and interacting with an external distribution called the background. The proposed methods are capable to preserve physical properties in the approximation of statistical moments of the problem like nonnegativity, entropy dissipation and asymptotic behaviour of the expected solution. The introduced methods are second order accurate in the transient regimes and high order for large times. We present applications of the developed schemes to the case of fixed and dynamic background distribution for models of collective behaviour.

Talk at the conference "Kinetic and Transport Equations: mathematical advances and applications", University of Parma.
A. Tosin, M. Z. Kinetic-controlled hydrodynamics for traffic models with driver-assist vehicles. Preprint arXiv: 1807.11476

The speed distribution of the vehicles in the traffic stream is at the core of the problem of road risk. Several reports on traffic safety in European and non-European countries stress that differences among the speeds of the vehicles are particularly responsible for a sensible increase in the crash risk. Not by chance modern Adaptive Cruise Control (ACC) technologies implemented in driver-assist and autonomous vehicles aim at adjusting automatically the vehicle speed so as to maintain a safe distance from the leading vehicle. In the transportation literature several studies are devoted to fit the statistical distribution of the speed of the vehicles out of experimental data. However, this approach can hardly lead to a model explaining how the speed distribution emerges from the individual behaviour of the drivers. In this talk we show that the Boltzmann-type kinetic theory, together with the asymptotic procedure of the grazing limit leading to a Fokker-Planck equation, can instead provide a simple yet effective explanation of the emergent speed distribution in traffic flow starting from a microscopic model of stochastic binary interactions among the drivers. The result is that the speed distribution can be well described by a Beta probability density function, whose parameters and statistical moments are explicitly related to the traffic density and to the parameters of the model of driver behaviour.

We develop a hierarchical description of traffic flow control by means of driver-assist vehicles aimed at the mitigation of speed-dependent road risk factors. Microscopic feedback control strategies are designed at the level of vehicle-to-vehicle interactions and then upscaled to the global flow via a kinetic approach based on a Boltzmann-type equation. Then first and second order hydrodynamic traffic models, which naturally embed the microscopic control strategies, are consistently derived from the kinetic-controlled framework via suitable closure methods. Several numerical examples illustrate the effectiveness of such a hierarchical approach at the various scales.

We develop a hierarchical description of traffic flow control by means of driver-assist vehicles aimed at the mitigation of speed-dependent road risk factors. Microscopic feedback control strategies are designed at the level of vehicle-to-vehicle interactions and then upscaled to the global flow via a kinetic approach based on a Boltzmann-type equation. Then first and second order hydrodynamic traffic models, which naturally embed the microscopic control strategies, are consistently derived from the kinetic-controlled framework via suitable closure methods. Several numerical examples illustrate the effectiveness of such a hierarchical approach at the various scales.

We present a semi-Lagrangian scheme for the approximation of a class of Hamilton-Jacobi-Bellman equations on networks. The scheme is explicit and stable under some technical conditions. We prove a convergence theorem and some error estimates. Additionally, the theoretical results are validated by numerical tests. Finally, we apply the scheme to simulate traffic flows modeling problems.

We present an analysis of risk levels on multi-lane roads. The aim is to use the crash metrics to understand which direction of the flow mainly influences the safety in traffic flow. In fact, on multi-lane highways interactions among vehicles occur also with lane changing and we show that they strongly affect the level of potential conflicts. In particular, in this study we consider the Time-To-Collision as risk metric and we use the experimental data collected on the A3 German highway.

Lane changing is one of the most common maneuvers on motorways. Although, macroscopic traffic models are well known for their suitability to describe fast moving crowded traffic, most of these models are generally developed in one dimensional framework, henceforth lane changing behavior is somehow neglected. In this paper, we propose a macroscopic model, which accounts for lane-changing behavior on motorway, based on a two-dimensional extension of the Aw and Rascle [Aw and Rascle, SIAM J.Appl.Math., 2000] and Zhang [Zhang, Transport.Res.B-Meth., 2002] macroscopic model for traffic flow. Under conditions, when lane changing maneuvers are no longer possible, the model "relaxes" to the one-dimensional Aw-Rascle-Zhang model. Following the same approach as in [Aw, Klar, Materne and Rascle, SIAM J.Appl.Math., 2002], we derive the two-dimensional macroscopic model through scaling of time discretization of a microscopic follow-the-leader model with driving direction. We provide a detailed analysis of the space-time discretization of the proposed macroscopic as well as an approximation of the solution to the associated Riemann problem. Furthermore, we illustrate some features of the proposed model through some numerical experiments.

This paper is concerned with mathematical modeling of intelligent systems, such as human crowds and animal groups. In particular, the focus is on the emergence of different self-organized patterns from nonlocality and anisotropy of the interactions among individuals. A mathematical technique by time-evolving measures is introduced to deal with both macroscopic and microscopic scales within a unified modeling framework. Then self-organization issues are investigated and numerically reproduced at the proper scale, according to the kind of agents under consideration.

In this paper we investigate the ability of some recently introduced discrete
kinetic models of vehicular traffic to catch, in their large time behavior,
typical features of theoretical fundamental diagrams. Specifically, we address
the so-called "spatially homogeneous problem" and, in the representative case
of an exploratory model, we study the qualitative properties of its solutions
for a generic number of discrete microstates. This includes, in particular,
asymptotic trends and equilibria, whence fundamental diagrams originate.

In this paper we propose a new modeling technique for vehicular traffic flow, designed for capturing at a macroscopic level some effects, due to the microscopic granularity of the flow of cars, which would be lost with a purely continuous approach. The starting point is a multiscale method for pedestrian modeling, recently introduced in Cristiani et al., Multiscale Model. Simul., 2011, in which measure-theoretic tools are used to manage the microscopic and the macroscopic scales under a unique framework. In the resulting coupled model the two scales coexist and share information, in the sense that the same system is simultaneously described from both a discrete (microscopic) and a continuous (macroscopic) perspective. This way it is possible to perform numerical simulations in which the single trajectories and the average density of the moving agents affect each other. Such a method is here revisited in order to deal with multi-population traffic flow on networks. For illustrative purposes, we focus on the simple case of the intersection of two roads. By exploiting one of the main features of the multiscale method, namely its dimension-independence, we treat one-dimensional roads and two-dimensional junctions in a natural way, without referring to classical network theory. Furthermore, thanks to the coupling between the microscopic and the macroscopic scales, we model the continuous flow of cars without losing the right amount of granularity, which characterizes the real physical system and triggers self-organization effects, such as, for example, the oscillatory patterns visible at jammed uncontrolled crossroads.

In this talk we analyse binary interaction schemes with uncertain parameters for a general class of Boltzmann-type equations with applications in classical gas and aggregation dynamics. We consider deterministic (i.e., a priori averaged) and stochastic kinetic models, corresponding to different ways of understanding the role of uncertainty in the dynamics, and discuss analogies and differences in the trends of some of their relevant thermodynamic quantities. Furthermore, via suitable scaling techniques such as the quasi-invariant limit, we derive the corresponding deterministic and stochastic Fokker-Planck equations in order to gain more detailed insights into the respective asymptotic distributions. We also provide numerical evidences of the trends estimated theoretically by resorting to recently introduced structure preserving uncertainty quantification methods. This is a joint work with M. Zanella (Politecnico di Torino).

We propose a class of optimal control problems for measure-valued nonlinear transport equations introduced in [2,3] to deal with many problems concerning vehicular traffic on networks. The objective is to minimise/maximise macroscopic quantities controlling few agents, for example smart traffic lights and automated cars. The measure theoretic approach allows to study in the same setting local and nonlocal drivers interactions and to consider microscopic control variables as additional measures interacting with the drivers distribution.

We consider a class of optimal control problems for measure-valued nonlinear transport equations describing traffic flow problems on networks. The objective is to minimise/maximise macroscopic quantities, such as traffic volume or average speed, controlling few agents, for example smart traffic lights and automated cars. The measure theoretic approach allows to study in a same setting local and nonlocal drivers interactions and to consider the control variables as additional measures interacting with the drivers distribution. We also propose a gradient descent adjoint-based optimization method, obtained by deriving first-order optimality conditions for the control problem, and we provide some numerical experiments in the case of smart traffic lights for a 2-1 junction.

In this paper we propose a classification of crowd models in built environments based on the assumed pedestrian ability to foresee the movements of other walkers. At the same time, we introduce a new family of macroscopic models, which make it possible to tune the degree of predictiveness of the individuals. By means of these models we describe both the natural behavior of pedestrians, i.e., their expected behavior according to their real limited predictive ability, and a target behavior, i.e., a particularly efficient behavior one would like they to assume (for, e.g., logistic or safety reasons). Then we tackle a challenging shape optimization problem, which consists in controlling the environment in such a way that the natural behavior is as close as possible to the target one, thereby inducing pedestrians to behave more rationally than what they would naturally do. We present numerical tests which elucidate the role of rational/predictive abilities and show some promising results about the shape optimization problem.

This paper presents a new approach to the modeling of vehicular traffic flows
on road networks based on kinetic equations. While in the literature the
problem has been extensively studied by means of macroscopic hydrodynamic
models, to date there are still not, to the authors' knowledge, contributions
tackling it from a genuine statistical mechanics point of view. Probably one of
the reasons is the higher technical complexity of kinetic traffic models,
further increased in case of several interconnected roads. Here such
difficulties of the theory are overcome by taking advantage of a discrete
structure of the space of microscopic states of the vehicles, which is also
significant in view of including the intrinsic microscopic granularity of the
system in the mesoscopic representation.

We consider a constrained hierarchical opinion dynamics in the case of leaders' competition and with complete information among leaders. Each leaders' group tries to drive the followers' opinion towards a desired state accordingly to a specific strategy. By using the Boltzmann-type control approach we analyze the best-reply strategy for each leaders' population. Derivation of the corresponding Fokker-Planck model permits to investigate the asymptotic behaviour of the solution. Heterogeneous followers populations are then considered where the effect of knowledge impacts the leaders' credibility and modifies the outcome of the leaders' competition.

In this work we focus on the construction of numerical schemes for the approximation of stochastic mean--field equations which preserve the nonnegativity of the solution. The method here developed makes use of a mean-field Monte Carlo method in the physical variables combined with a generalized Polynomial Chaos (gPC) expansion in the random space. In contrast to a direct application of stochastic-Galerkin methods, which are highly accurate but lead to the loss of positivity, the proposed schemes are capable to achieve high accuracy in the random space without loosing nonnegativity of the solution. Several applications of the schemes to mean-field models of collective behavior are reported.

In this talk we present a Boltzmann-type kinetic approach to the modelling of road traffic, which includes control strategies at the level of microscopic binary interactions aimed at the mitigation of speed-dependent road risk factors. Such a description is meant to mimic a system of driver-assist vehicles, which by responding locally to the actions of their drivers can impact on the large-scale traffic dynamics, including those related to the collective road risk and safety.

In this work we present a two-dimensional kinetic traffic model which takes into account speed changes both when vehicles interact along the road lanes and when they change lane. Assuming that lane changes are less frequent than interactions along the same lane and considering that their mathematical description can be done up to some uncertainty in the model parameters, we derive a hybrid stochastic Fokker-Planck-Boltzmann equation in the quasi-invariant interaction limit. By means of suitable numerical methods, precisely structure preserving and direct Monte Carlo schemes, we use this equation to compute theoretical speed-density diagrams of traffic both along and across the lanes, including estimates of the data dispersion, and validate them against real data.

In this talk we will consider two classes of differential models for crowd dynamics: the discrete (i.e., agent-based) ones, which are formulated by means of ODEs, and the continuous (i.e., density-based) ones, which are formulated by means of PDEs. Since different descriptions can produce different observable results, it is a priori questionable what is the most suitable mathematical representation. On the other hand, independently of the representation, models are often based on common phenomenological assumptions, therefore they are expected to describe analogous phenomena. Typical procedures relate discrete and continuous models by sending the total number N of the agents to infinity while keeping the total mass of the system constant. This implies that the mass of the single agents becomes infinitesimal when N grows, so that the continuous description ”emerges” in the limit from the discrete one. Conversely, in this talk we will consider continuous models per se, parallelly to discrete ones, for any number of pedestrians. This way agents can be thought of a priori as massive, i.e., having a constant finite (say unitary) physical mass, but the total mass of the system grows with N. We will discuss in which sense the two types of models can be compared from this point of view and how they can be embedded in a multiscale perspective.

In this seminar we present a Boltzmann-type kinetic approach to the description of the interplay between vehicle dynamics and safety aspects in vehicular traffic. Sticking to the idea that the macroscopic characteristics of the traffic flow, including the distribution of a properly defined concept of driving risk, are generated by one-to-one interactions among drivers, we discuss a model which links the personal (i.e., individual) risk to the changes of speeds of single vehicles and implements a probabilistic description of such microscopic interactions in a Boltzmann-type collisional operator. By means of suitable statistical moments of the kinetic distribution function, this approach makes it possible to recover asymptotic relationships between the average risk and the road congestion, which show an interesting and reasonable correlation with the well-known free and congested phases of the flow of vehicles.

We consider a collision avoidance mechanism between pedestrians based on sidestepping: when a pedestrian estimates that s/he is going to collide with a neighbouring walker s/he deviates leftwards or rightwards by a certain angle. S/he evaluates dynamically the probability to collide by assessing the time to collision, i.e. an extrapolation of the time needed to get closer to the other pedestrian than a certain given distance. As a complementary action, s/he also tries to walk in a desired direction identified by another angle. These minimal microscopic rules lead to complex emergent macroscopic phenomena, such as the alignment of the velocity in unidirectional flows and the lane or stripe formation in bidirectional flows. We discuss the collision avoidance mechanism at the microscopic scale, then we study the corresponding Boltzmann-type kinetic description and its mean-field approximation in the quasi-invariant direction limit. In the spatially homogeneous case we prove directional alignment under specific conditions on the sidestepping rules for both the collisional and the mean-field model. In the spatially inhomogeneous case we illustrate numerically the rich dynamics that the proposed model is able to reproduce.

In this talk I will present a kinetic approach to the modeling of vehicular traffic. Sticking to the idea that the macroscopic characteristics of traffic flow are ultimately generated by one-to-one interactions among drivers, the approach consists in linking the changes of speed of single vehicles to binary interactions among drivers, implementing then a probabilistic description of such microscopic interactions in a Boltzmann-type collisional operator. In particular, I will discuss how this approach allows one to study the fundamental diagrams of traffic, possibly considering a heterogeneous composition of the flow of vehicles, the traffic flow on single roads as well as on road networks, up to some preliminary ideas on traffic safety issues.

In this talk we present a modelling approach to multi-agent systems, especially vehicular traffic and human crowds, based on Boltzmann-type kinetic equations and measure-valued conservation laws. We discuss how to pass from microscopic binary interactions to the description of emerging collective trends by means of probabilistic and multiscale representations of the particle system. In particular, we show that hyperbolic equations with non-local flux accounting for interactive dynamics arise naturally in such a context.

In this paper we present a Boltzmann-type kinetic approach to the modelling of road traffic, which includes control strategies at the level of microscopic binary interactions aimed at the mitigation of speed-dependent road risk factors. Such a description is meant to mimic a system of driver-assist vehicles, which by responding locally to the actions of their drivers can impact on the large-scale traffic dynamics, including those related to the collective road risk and safety.

We study a nonlinear transport equation defined on an oriented network where the velocity field depends not only on the state variable, but also on the solution itself. We prove existence, uniqueness and continuous dependence results for the solution of the problem intended in a suitable measure-theoretic sense. We also provide a representation formula in terms of the push-forward of the initial and boundary data along the network and discuss an example of nonlocal velocity field fitting our framework.

In this paper we study binary interaction schemes with uncertain parameters for a general class of Boltzmann-type equations with applications in classical gas and aggregation dynamics. We consider deterministic (i.e., a priori averaged) and stochastic kinetic models, corresponding to different ways of understanding the role of uncertainty in the system dynamics, and compare some thermodynamic quantities of interest, such as the mean and the energy, which characterise the asymptotic trends. Furthermore, via suitable scaling techniques we derive the corresponding deterministic and stochastic Fokker-Planck equations in order to gain more detailed insights into the respective asymptotic distributions. We also provide numerical evidences of the trends estimated theoretically by resorting to recently introduced structure preserving uncertainty quantification methods.

The purpose of this paper is to study the properties of kinetic models for traffic flow described by a Boltzmann-type approach and based a continuous space of microscopic velocities. In our models, the particular structure of the collision kernel allows one to find the analytical expression of a class of steady-state distributions, which are characterized by being supported on a quantized space of microscopic speeds. The number of these velocities is determined by a physical parameter describing the typical acceleration of a vehicle and the uniqueness of this class of solutions is supported by numerical investigations. This shows that it is possible to have the full richness of a kinetic approach with the simplicity of a space of microscopic velocities characterized by a small number of modes and in this case, the explicit expression of the asymptotic distribution paves the way to deriving new macroscopic models using the closure provided by kinetics.

Experimental studies on vehicular traffic provide data on quantities like density, flux, and mean speed of the vehicles. However, the diagrams relating these variables (the fundamental and speed diagrams) show some peculiarities not yet fully reproduced nor explained by mathematical models. In this paper, resting on the methods of kinetic theory, we introduce a new traffic model which takes into account the heterogeneous nature of the flow of vehicles along a road. In more detail, the model considers traffic as a mixture of two or more populations of vehicles (e.g., cars and trucks) with different microscopic characteristics, in particular different lengths and/or maximum speeds. With this approach we gain some insights into the scattering of the data in the regime of congested traffic clearly shown by actual measurements.